Former Fed chair Ben Bernanke argued in January that an audit raises the risk that "reviews and recommendations concerning individual monetary policy decisions would provide a vehicle for members of Congress to apply political pressure on the Fed."

But Trump actually has already accused the Fed of being too political.

In an interview with The Hill in October, Trump said Yellen was keeping interest rates low so President Obama "doesn't want to have a recession-slash-depression during his administration."

That interview took place before the Fed raised rates for the first time in nine years in December. But rates remain extremely low.

Trump has repeatedly expressed concerns that the stock market is a bubble waiting to burst because of Fed policies.

He told CNBC in September 2012 that Bernanke's quantitative easing program of buying mortgage-backed securities was leading to "phony numbers" in the economy.

Presumably, a Democratic president would renominate Yellen -- assuming she still wants the job.

But presidents have historically opted for continuity at the central bank -- regardless of whether or not their political views align with the Fed chair.

Ronald Reagan reappointed Paul Volcker as Fed chair even though Volcker was originally chosen by Jimmy Carter. Bernanke was nominated for a second term as Fed chair by Obama, after first serving under George W. Bush.

There have been only seven Fed chairs since 1951. It seems that being a Supreme Court justice or coach of the Pittsburgh Steelers are the only professions that offer greater job stability.

So it would be somewhat unprecedented for a President Trump to make drastic changes at the Fed.

Interestingly though, Trump seems to have a soft spot for Volcker -- who raised interest rates aggressively in the 1980s to fight inflation.

Trump told Bloomberg in August that even though he always loved low interest rates as a real estate developer, he thought that Volcker was a "terrific guy" who was "doing what had to be done."

Could Volcker be back as Fed chair if Trump makes it to the Oval Office? That seems unlikely.

But based on what Trump has said about Yellen, it seems he would want someone who is willing to hike interest rates far more dramatically than the current Fed.